CaRBM: A Fixed-Depth Quantum Algorithm with Partial Correction for Thermal State Preparation

We introduce the CaRBM algorithm for fixed-depth thermal state preparation. Our algorithm is based on thermal state purification and uses a block-enchoding scheme based on a Restricted Boltzmann Machine (RBM) architecture to implement the imaginary-time propagator e^-βH, which is implemented in the quantum circuit in a fixed-depth manner via Cartan decomposition. The non-unitary evolution is implemented in a probabilistic manner by coupling to an ancillary qubit and performing post-selection on the results. The Cartan decomposition circuit implementation allows us to correct several layers of the non-unitary evolution, which guarantees the success of the post-selection process on these layers, thereby significantly improving the overall acceptance rate of the entire evolution. We show that we can correct as many layers as the number of physical qubits. We demonstrate our algorithm by running circuit simulations to find the partition function zeros of the XXZ model and the phase diagram of the Gross-Neveu model, which is a model of strongly interacting relativistic fermions.